In fruit storage, new methods are being sought to extend the distribution period while maintaining the highest quality parameters of the fruit, i.e., the physical and chemical characteristics of the fruit, but also the health-promoting properties. One method is to treat the fruit with 1-MCP, which effectively inhibits fruit ripening, since the main reasons for limiting the distribution of minikiwi fruit are rapid ripening and the loss of firmness. It is also highlighted that minikiwi is a source of antioxidants, which, as highly reactive compounds, are quickly degraded during storage. This study evaluates the effectiveness of using 1-MCP to reduce minikiwi softening and maintain the high antioxidant properties of the fruit. In the experiment, minikiwi fruits of the ‘Ananasnaya’ cultivar were used. After harvesting, the fruits were treated with 1-methylcyclopropene at a concentration of 0.65 µL/L. Fruits treated in this way were stored in ordinary cold storage (NA) and under low oxygen concentration (ULO) conditions for a period of 12 weeks. The fruits’ physicochemical properties, sugar and acid contents and antioxidant potential, which consisted of ascorbic acid, polyphenols, phenolic acids, flavonols and flavan-3-ols, were evaluated. The application of 1-MCP is effective in reducing the loss of firmness in the minikiwi during storage, but the blocking of minikiwi ripening by 1-MCP makes the fruit less sweet and more acidic. Fruits treated with 1-MCP had higher antioxidant activity and a higher content of biologically active compounds. The effect of 1-MCP is stronger for flavan-3-ols, but slightly weaker for phenolic acids. Fruits treated with 1-MCP have a higher antioxidant potential than untreated fruit after a long period of storage.